We already have at least one true baby-swap. Gilly's child and Mance's. We already have Sam telling us how Maesters know to give babies goat's milk in lieu of mother's milk. We already know that even those closest to Gilly and Mance's babies couldn't tell that they'd been switched.
We already know that the male children were the heirs. We already know that Aegon was more important, and in line before, Rhaenys and Viserys. We already know Ned brought up Viserys before knowing another existed.
We already know that Elia was aware of Aegon's importance since the time of his birth. We already know the KG were sworn to protect the King. We already know Rhaegar was never King, but Aegon was.
We already know the KG at the ToJ did not consider Viserys the King.
We already know that Lyanna may have given birth within days to a few weeks of the ToJ battle.
We already know there is little explanation why the KG would hang around a pregnant woman, who may or may not give birth to a live healthy male heir.
We already know Rhaegar had some powers of prophecy.
We already know that both Gerold and Ned found the ToJ through someone.
We already know it was Maester Walys who brokered the Stark marriages. We already know the Citadel has a hand in everything.
We already know Aegon was the true heir after Aerys and Rhaegar died. We do NOT know that Lyanna's supposedly male was the true heir.
We already know Ned was able to lie to someone on their deathbed re: taking care of their children.
We already know Ned has brought up the question of what a parent would do when it comes to their children.
We already know everything.
Rhaegar brought Aegon south about the time that R disappeared from KL, with Elia's consent, to hide him, along with two KG, because they believed he was TPTWP. To DORNE, Elia's home.
They switched him with the closest Targ-looking baby available - Ashara Dayne's. Ashara, handmaid to Elia, connected romantically to at least one Stark, possibly impregnated around the same time that Elia was impregnated with Aegon, sister to a KG, and not nailed down to Starfall.
They had no reason to believe Elia and children would be killed. They only cared to try and hide TPTWP, the Song of Ice and Fire.
Rhaenys was never in line for the throne. Elia stayed in KL as a prisoner wither her daughter well after her son Aegon was spirited away. Her supposed son, a Dayne, "looked like a Targ" with his violet eyes.
When Aerys ordered Gerold to find Rhaegar, Gerold got the information where to find them from his bastard relative, Maester Walys Flowers, Lyanna's Maester since she was born. Needle in a haystack he found them... and stayed there instead of returning to his King, or fighting with his Prince at the Trident.
At the Sack, Rhaenys was scared and ran away from her bed. Or her mother told her to run and hide. Or her mother was assured by someone else that Rhaenys was OK. Regardless, Elia kept an innocent baby in her arms out of motherly instinct, knowing her son was far away.
Best case scenario for Elia would be that she, Rhaenys and the babe would be banished to Dragonstone to join Rhaella and Viserys. Worst case, as it happened, would be that the babe was killed... with the plus side being everyone would think Aegon was dead, and he was therefore safe.
Then, someone told Ned where to find Lyanna, another needle in a haystack. Again, the culprit is Walys, Maester of his House, son of an Archmaester, whose fate is mysteriously unknown.
Conclusion: Varys swapped the wrong child, but believes the child to be Aegon. fAegon is a Dayne/Stark.
GRRM already gave us Robert's deathbed scene, with all the discussion of killing Targ babies and Ned floundering to express a promise, and the whole scenario between Gilly/Mance's babe at the Wall, and how Maesters know how to keep a babe alive without a wetnurse or mother. Why isn't anyone listening?